Mathematics for the interested outsider

The Branching Rule, Part 1

That is, we want to replace these numbers with objects of a category, replace the sum with a direct sum, and replace the equation with a natural isomorphism.

It should be clear that an obvious choice for the objects is to replace with the Specht module , since we’ve seen that . But what category are they in? On the left side, is an -module, but on the right side all the are -modules. Our solution is to restrict , suggesting the isomorphism

This tells us what happens to any of the Specht modules as we restrict it to a smaller symmetric group. As a side note, it doesn’t really matter which we use, since they’re all conjugate to each other inside . So we’ll just use the one that permutes all the numbers but .

Anyway, say the inner corners of occur in the rows , and of course they must occur at the ends of these rows. For each one, we’ll write for the partition that comes from removing that inner corner. Similarly, if is a standard tableau with in the th row, we write for the (standard) tableau with removed. And the same goes for the standard tabloids and .

Our method will be to find a tower of subspaces

so that at each step we have as -modules. Then we can see that

And similarly we find , and step by step we go until we find the proposed isomorphism. The construction itself will be presented next time.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.